Boolean Algebra and Logic Gates

What is Boolean Algebra?

In 1854, George Boole, a 19^th century English Mathematician has invented Boolean/ Logical/ Binary algebra by which reasoning can be expressed mathematically.

Boolean algebra is one of the branches of algebra which performs operations using variables that can take the values of binary numbers i.e., 0 (OFF/False) or 1 (ON/True) to analyze, simplify and represent the logical levels of the digital/ logical circuits.

0<1, i.e., the logical symbol 1 is greater than the logical symbol 0.

Boolean Algebra Formula

Following are the operations of Boolean algebra:

1. OR Operation
2. AND Operation
3. Not Operation

OR Operation

The symbol ‘+’ denotes the OR operator.

To perform this operation we need a minimum of 2 input variables that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

OR operation is defined for A OR B or A+B as if A = B = 0 then A+B = 0 or else A+B = 1.

The result of an OR operation is equal to the input variable with the greatest value.

Following are the possible outputs with a minimum of 2 input combinations:

• 0 + 0 = 0
• 0 + 1 =1
• 1 + 0 = 1
• 1 + 1 = 1  

AND Operation

The symbol ‘.’ denotes the AND operator.

To perform this operation we need a minimum of 2 input variables that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

And operation is defined for A AND B or A.B, if A = B = 1 then A.B = 1 or else A.B = 0.

The result of an AND operation is equal to the input variable with the lowest value.

Following are the possible outputs with a minimum of 2 input combinations:

• 0 .0= 0
• 0 .1 = 0
• 1 .0 = 0
• 1 .1 = 1

Not Operation

Not operation is also known as Complement Operation. This is a special operation which is denoted by ‘’’. Complement of A is represented as A’.

To perform this operation we need a minimum of 1 input variable that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

Not operation is defined for A’ or NOT A if A = 1, then A’ = 0 or else A’ = 1.

The result of a not operation is the inverse value of the set of inputs provided.

Following are the possible outputs with minimum 1 input value:

• (1)’ = 0
• (0)’ = 1

Boolean Algebra Rules

The Following are the important rules followed in Boolean algebra.

• Input variables used in Boolean algebra can take the values of binary numbers i.e., 0 or 1. Binary number 1 is for HIGH and Binary 0 is for LOW.
  
  
• The complement/negation/inverse of a variable is represented by ‘
  Thus, the complement of variable A is represented as A’. Thus A’ if A = 1, then A’ = 0 or else A’ = 1
  
  
• OR-ing of the variables is represented by a ‘+’ sign between them. For example, OR-ing of A, B is represented as A + B.
  
  
• Logical AND-ing of two or more variables is represented by a ‘.’ sign between them, such as A.B. Sometime the ‘.’ may be omitted like AB.

Boolean Laws

1. Commutative law
2. Associative law
3. Distributive law
4. AND law
5. OR law

Commutative law

Binary operation(s) satisfying anyone of the following expressions is/are said to a commutative operation.

• A.B = B.A
• A+B = B+A

Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

Associative law

Law of Associative states that the order in which the logic operations are performed is irrelevant as their effect is the same.

• (A.B).C = A.(B.C)
• (A+B)+C = A+(B+C)

Distributive law

Law of Distributive states the following conditions

1. A.(B+C) = A.B + A.C
2. A+(B.C) = (A+B).(A+C)

AND law

AND Law states the following conditions as they are using AND Operations.

A.0 = 0
A.1 = A
A.A = A
A.A’ = 0

OR law

OR laws states the following conditions as they are using OR Operations.

A+0 = A
A+A = A
A+1 = 1
A+A’ = A

Complement law

This law uses the NOT operation. The law of Complement also known as Inversion/Negation, states that double inversion of a variable result in the original variable itself.

• (A’)’ = A or A + (A’)’ = 1

Boolean Algebra Truth Table

Truth Table of AND Operation and OR Operation

A	B	A+B	A.B
0	0	0	0
0	1	1	0
1	0	1	0
1	1	1	1

Truth Table of NOT operation

A	A’
0	1
1	0

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Logic Gates

Digital systems are said to be built using Logic Gates. A Logic gate is an electronic circuit or logic circuit which can take one or more than one input to get only one output. A particular logic is the relationship between the inputs and the output of a logic gate.

Types of Logic gates

1. AND Gate
2. OR Gate
3. NOT Gate
4. NAND Gate
5. NOR Gate
6. XOR Gate
7. XNOR Gate

AND Gate

This logic gate uses AND operation logic and denoted by

Input (A)	Input (B)	Output
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate

This logic gate uses OR operation logic and denoted by

A	B	Output
0	0	0
0	1	1
1	0	1
1	1	1

NOT Gate

This logic gate uses NOT operation logic & denoted by

It is also known as an Inverter.

Input	Output
0	1
1	0

NAND Gate

A NOT-AND operation is known as NAND operation, and a logic gate using this NAND operation logic is called NAND gate.

Here the output of AND gate is the input of the NOT gate and the output of this combination of NOT gate and AND gate is the output of the NAND gate.

NAND Gate Diagram

NOR Gate

A NOT-OR operation is known as NOR operation, and a logic gate using this NOR operation logic is called the NOR gate.

Here the output of the OR gate is the input of the NOT gate and the output of this combination of NOT gate and OR gate is the output of the NOR gate.

NOR Gate Diagram

Input (A)	Input (B)	Output (A ⊕ B)
0	0	1
0	1	0
1	0	0
1	1	0

XOR Gate

XOR or EXOR or Exclusive-OR is a special type of gate or circuit which will give high output if even or zero number of inputs are high or else it will give low output.

The algebraic expressions   and  both represent the XOR gate with inputs A and B.

The Operation of this gate is denoted by ⊕

XOR Gate Logic

D = A XOR B
D = A ⊕ B      
D = A’.B + A.B’

XOR Logic Diagram

XOR Truth Table

Input (A)	Input (B)	Output (A ⊕ B)
0	0	0
0	1	1
1	0	1
1	1	0

XNOR Gate

XNOR or EX-NOR or Exclusive NOR gate is a special type of gate or circuit that will give high output if an odd number of inputs are high or else it will give low output. It is the opposite of the XOR gate.

The Operation of this gate is denoted by ‘Ɵ’.

XNOR Gate Logic

D = A Ɵ B
D = A’.B’ + A.B

XNOR Gate Diagram

XNOR Truth Table

Input (A)	Input (B)	Output (A Ɵ B)
0	0	1
0	1	0
1	0	0
1	1	1